Upper Bound on the Number of C ycles in B order-deci sive C ellu lar A utomata

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1 Com plex Sys tems 1 (1987) Upper Bound on the Number of C ycles in B order-deci sive C ellu lar A utomata P uhua Guan Department of Matbematics, University of Puerto Rico, Rio Piedras, PR 00931, USA YuHe Center (or Com plex Systems Research and Department of Physics University of Illinois, 508 South Six th S treet, Champaign, , USA Abstract. The number of stable states of anyone-dimensional k state border-decisive cellu lar automaton on a finite lattice with periodic boundary conditions is proved to be bounded by k R and the number of cycles of length p is bounded by ~ L q!pjl(p/ q)k4 R, where (R + 1) is the number of neighbors and }J is the Mobius function. 1. Int roduction We consider a deterministic cellular automaton (CA) [1] on a one-dimensional array of N sites. At each site i, there is a dynamical variable whose value at t ime t will be den oted by a~t). Here a~t) be longs to a finite set S with k elements. The state of the CA with N sites can be described by an N component vector.04(1) (.04(') E SN), whose the ith-eomponent is aill. The time evolution is given by A(t+l) = 4>(A(t)) where 4> is defined by a local deterministic ru le 4>: sr+l -- S of the form wh ere rl an d r2 (l.1) are the nu mb er of neighbors to the left and to the right respectively, and r l + r 2+1 (:;:;; R + 1) is the to tal number of neighbors. The t ime evolution of a CA on a finite lat t ice can be represented by a graph, called the state transition diagram, whose nodes correspond to the states of the CAj directed arcs connecting the nodes represent transitions between the states. In general, the complete topology of the state transition diagram is difficult to determine except for a special class of CA, vls., additive CA [2,3]. In t his note we give upper bou nds for the number of stable states and of cycles of a more general class of CA, border-decisive CA Comp lex Systems P ublications, Inc.

2 182 Puhua Guan and u He D efinition 1. A one-dimension al N -site k -state deterministic CA is said to be border-decisive if its local rule 4> sa tisfies (ai - r Il ~-r l+ ll...,a.; +r2) =I=-.p(Cli-rp a;-rl+h..., a~+"2 ) if ai+ r2:f= a~ +r2 (1.2) with T2 > a and for any fixed values of (a;- rl' 0..;-"1+1,..., a; +r2-1), or 4J(ai-rl' Q;-rl+h "'J Q;+"2) 'I (a~_rl' ai - rl+h..., lzt+r,) if ai-rt '# a~ _" l (1.3) with Tl > a and f or any fixed values of (tlt-rl+1j lls"- rl+%'..., lli+r2)' Note that we impose period ic boundary condit ions and hence, the site indices are computed modulo N. In words, for a border-decisive CA rule, if all the neighbors of site i except the one on th e "border" (leftmost or rightmost) are fixed at time t, then ajt+l) will be different for different values of the border site at t. The class of border-decisive CA has been investigated 141 also und er the name of toggled rule 151and left and right permutive rule Results Theorem 1. The number of stable states of any border-decisive CA has an upper bound nl = k R Proof. Let q, be a CA rule satisfying (1.2). (The same proofgoes through for ru les satisfying (1.3).) For any fixed (a.-r pa.-rl +h...,a.+r1-d, 4h: S --+ S defined by (2.1) is a one-to-one function. Since S is finite, 4Jl is also onto and, therefore, 4Jl is inver t ible. Now suppose A is a stable state, i.e., ~ (A ) = A. The values of a string of R sites (a.-rt' a.-rl+h..., aj+r:a-d for arbitrary i determine the entire set of stable st ates as follows: For arbitrary but fixed value of this string we can compute (l;"+r1 from _..(- 1)( ) a.+r1 - If'l Bj. (2.2) Knowing Bi+r:a, 4; +r1+l can be determined, and so on. Since we have imposed periodic boundary conditions we event ually return to the original string of R sites. A st able state ensues if and only if the values thus obtained are identical to those assigned initially to the string. Since there are k R choices for the va lues of the st ring the number of stable states cannot exceed k R. T heorem 2. The number of cycles of length p of any border-decisive CA bss an upper bound rip = (k PR - ~qlp,q~pqnq )/p = ~ ~qlp!l ( p/ q)kqr.

Num ber of Cycles in Border-Decisive Cellular Automata 183 Proof. Consider a state.ii on a cycle of length p, i.e., <I>P(.ii) =.ii. Since 4>, (which is assumed to satisfy (1.

3 Num ber of Cycles in Border-Decisive Cellular Automata 183 Proof. Consider a state.ii on a cycle of length p, i.e., <I>P(.ii) =.ii. Since 4>, (which is assumed to satisfy (1.2)) is invertible, the values of a rectangular R h f h. [ (I) (t) I r X P patc 0 t e space-time pattern 4 i - r 1,..., 4 i + r :l - 1 lor t - 1,2,..., p and arbitrary i determine the entire set of states on cycles of lengt h p. First a!2 r 2 for t = 1, 2,..., P can be determined uniquely by successively using (,) _.1.(- 1) 1 (1+1)1 a i + r 2 - 'f'l a i. (2.3) (Not e that a\'+p) = a\') for all j and t due to the pe riodic ity in time.) Then th e values of the rest of t he sites can be ascertained by iterat ing the above procedure. As before, the periodic bound ary conditions lead to consiste ncy requirements. Since the initial patch can assume k pr different configurations, one eas ily obtains the weaker bound k pr / p. (If b is a bo und for the number of states on cycles of length p, then b/ p is the corresponding bound for the number of cycles of lengt h p.) However, some of the k pr configurations for the initial patch have period q «p) where q I P (q is a divisor of pl. Clearly, for such configurations of the initial patch if the consistency cond itions are satisfied one obt ains a cycle of length q because the periodicity of the initial patch guarantees a!2 r 2 = a~~~:) for t = 1.2,..., q. If the consistency cond itions are not satisfied one does not obtain a cycle of lengt h either p or q. So we can, in fact, subtract the number of such configurations (which can be expressed in te rms of n q ) from k pr and thus obtain the maxi mum number of states which can be on cycles of lengt h Pi dividing by p yields the upper bound for the number of cycles. If we write the bound iterat ively it is easy to see th at the first expression for the upper bound in Claim 2 is obtained. Alternatively, the bound can also be expressed in terms of the Mobius function /lo, where /lo(i) = 1; /lo(n) = 0 if n has a squared fact or; and "(PIP""P,) = (-1)" if all the primes Ph P"..., PI are different [71 T his upper bound on the number of cycles of a given length depends only on k and R. It is independent of the size N of the CA as long as N is finite. For k = 2, R = 2, the bounds on number of cycles of length s up to ten are 4, 6, 20, 60, 204, 670, 2340, 8160, 29120, and respect ively. In addit ion, it is easy to see that the following statements are true: Let N be the smallest size of a given border-decisive CA such that t he upper bound n p for the number of cycles of length P is reached (this is possible only if pn p ~ k N ) ; then the same CA will reach the same bound for size M > N if and only if M is a mult iple of N. (Note that there is no rest riction on M being finite.) On the other hand, if the actua l number of cycles of length p is m < n p for a CA with N sites, th en t he number of cycles of length p for the same CA of size M is no less than m when M is a multiple of N. Thus by calculat ing the number of cycles for a CA for relat ively sma ll sizes one can get the lower bound for, and somet imes even the exact value of, th e number of cycles for very large sizes. (This is possib le only for sizes wh ich have relatively small divisors.) For example, for Ru le 90 when N = 3 there are four I-cycles. We can immediately deduce that all sizes N which

184 Puhua Guan and Yu He are multiples of 3 have four f-cycles. For N S 30 the maximum number of cycles of length p for p :0; 8 are: 4, 6, 20, 60, 51, 670, 36, and 8160 [21.

4 184 Puhua Guan and Yu He are multiples of 3 have four f-cycles. For N S 30 the maximum number of cycles of length p for p :0; 8 are: 4, 6, 20, 60, 51, 670, 36, and 8160 [21. These first occur for N = 3, 6, 15, 12, 17, 30, 9, and 24 respectively and hence, the upper bound is reached for p = 1, 2, 3, 4, 6, 8. However, for Rule 30 and N ~ 17 the corresponding maximum number of cycles are: 3, 0, 4, 7, 4, 0, 15, 1 [8]. We will now determine, for a given k and R, how many of the kkll+ 1 possible CA rules are border-decisive. Let B 1 be the set of rules satisfying (1.2), B, that satisfying (1.3), and B be the set of all border-decisive rules. We have B = B 1 UB, and IBI, the cardinality of B, satisfies IBd :0; IBI = 21Bd - IB, n B,I :0; 21Bd (2.4) where we have used the symmetry relation IBII = IB 2 1. It is easy t o compute IBd Notice that for fixed ("<-,,, "<-,, +1,...,,,<+,,- 1), 1>1: S ~ S defined by (2.1) is actually a permutation of S, i.e., <PI is an element of the permutation group of k elements Sk' Since a border-decisive rule is specified by assigning a 4>1 to each value of (ai- rl, 4i - rl+l,..., 4i+r2- d E SR, it is a mapping from SR t o Sk ' On the other hand, it is obvious that each mapping from S R to 51; gives a border-decisive CA rule. From ISkl = kl and ISRI = k R, we have IB 11 = (ki). (2.5) Compar ing with the total number of possible CA rules, R = kk R + 1, we have, (2.6) In t he cases one usually cons iders, where both k and R are small, the fraction of border- decisive rules is not neg ligible. For small k, IB 1 n B 2 1 can be easily calculated. IB, n B,I = 2,R-' for k = 2 and IB, n B,I = 12,R-' for k = 3. Given IB 1 n B,I and IB 1 1, the number of border-decisive rules IBI can be computed from (2.4). For k = 2, R = 1, IBI= 6 (JP = ), for k = 2, R = 2, IBI = 28 (JP '" 0.109), for k = 3, R = 1, IBI = 420 (JP '" 0.021). 3. Some generalizations T he res ults of the last section can be extended to the in-degree of any node [i.e., the number of predecessors) of the state transition diagram and to higher-o rder rules. We state the following corollaries without further explanation since the proofs are similar to the ones given above. Corollary 1. The in-degree of any node of the state transition diagram of a border-decisive CA with k states and R + 1 neighbors is bounded above by k R

5 N um ber of Cycles in Border-Decisive Cellular A utomata 185 Definition 2. A deterministic CA rule ~ of order s is given by (3.1) where <l! is defined by the local rule 4>: S A+! ~ S of the form Here R + 1 = L:~I (TI: + T,: + 1). A deterministic CA rule of ord er s on a finite lattice is border-decisive if the function <p in (3.2) is a one-to-one, function of the argument a~~~~: l ), where T2 j ( > 0» T 2; for all i :f: i, for each set of fixed values of the other R arguments, or if <P is a one-to-one function of the argument a~:~:: l ) I where T l j (> 0» Tl; for all i :f: i, for each set of, fixed values of the other R arguments. Let R = r r, + T2,t where r r, = maxi(tl;) and T2,t = maxi(t2;} ' We have. then: Corollary 2. Th enumber ofstable states ofa finite k -state s-ord er borderdecisive CA is bounded above by nl = k R and the number ofcycles ofleng th p is bounded above by np = (k pr - L,!p,, #p qn, )/ p = ~ L,lpJ1.(p/q)k,R. A cknow ledgement We thank Professors C. Jayaprakash and S. Wolfram for useful suggest ions and a reading of the manuscript. Yu He ack nowledges support from the NSF under Grant No, DMR in the initial stages of t he work, and parti al support from the National Center for Supercomputing Applications in the lat ter stages. Refe rences [I) S. Wolfram. "St at istical Mechanics of Cellular Automata", Reviews of Modern Phys ics 55 (1983) !2j O. Martin, A. M. Od lyzko, and S. Wolfram, "Algebraic Properties of Cellular Automata", Communications in Mathematical Physics 93 (1984) (3] P. Gu an and Y. He, "Exact Resu lts for Deterministic Cellular Automata with Addi tive Rules", J ournal of Statistical Physics 43 (1986) [4J S. Wolfram, "Computat ion Theory of Cellular Au tomata", Ccmmunicaticns in Mathematiclli Physics 9 6 (1984) [5] H. Gu towitz, J. Victor, and B. Knight, "Local Structure Theory of Cellular Automata", (preprint).

6 186 Puhua Guan and Yu He [6] G. Hed lund, "Endomorphism and Automorphism of the Shift Dynamical System" I Mathematical Systems Theory 3 (1969) and J. Milnor. "Some Notes on Cellular Automaton-Maps" I (preprint). [7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, (Oxford, 1979). 18] S. Wolfram, "Random Seque nce Generation by Cellular Automata", Advances in Applied Mat hematics 7 (1986)

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